Where does the curvature of space happen? I've never understood the curvature of space. Does the action of curving travel through space itself?
I've asked this a lot and no one gave me an answer and I'm still looking for an answer, or general relativity is wrong.
I can't understand the curvature of space, because curvature of something happens in space while curvature of space needs space itself to happen, so it's contradictory. It is like saying water can get wet, or fire can get burned. So where does the curvature of space described in general relativity happen? Does space curve "inside" itself?
 A: *

*General Relativity is a model of reality. It is not reality: reality lies in the physical phenomena.


*That said, GR is a highly successful model: many experiments and observations conform to its predictions.


*GR is a model of intrinsic curvature: no extrinsic curvature is required.
If you wish to explore whether the GR model could be enhanced by considering extrinsic curvature, you must propose an experiment or observation of some phenomenon affected by extrinsic curvature. This seems impossible, since all experiments and observations take place within the universe that GR models. We cannot observe the universe from outside.
A: The surface of the Earth can be mapped using longitudes and latitudes, and plotted as a world map on a flat paper. For example, a US state basically limited by parallels and meridians as Kansas can be plotted with plane polar coordinates, where the parallels are circles and meridians are straight lines. However, the distance between cities using a ruler and a fixed appropriate scale factor are slightly different from the known measured distances.
We know the reason: a spherical surface can not be mapped on a flat surface without distortion. But once we construct formulas to correct the ruler distances, the intuition of viewing the planet from outside is no longer necessary.
A similar situation happens when a far away observer maps the space around a planet using spherical coordinates. The distances between points on the map are slightly different from what is locally measured. But in this case, the formulas to correct them can not be visualized as a "curved space" viewed from outside, because there is no outside. Nevertheless the maths machinery have the same grounding as the used for the previous example, and the geometric expression are stretched to describe higher dimensions than surfaces.
A: Let's start with curved $2$D space. You have certain expectations about geometry in a flat plane.

*

*Two parallel lines never meet.

*The angles of a triangle add to $180^o$.

*If you walk around a square, you come back to your starting point.

*The distance between two points $(x_0,y_0)$ and $(x_1,y_1)$ is given by Pythagorus' theorem. $d = \sqrt{\Delta x^2 + \Delta y^2}$.

The surface of the earth looks flat. One way to tell that it is curved is that these expectations are violated. On a small scale, the violations are so small you can't tell the difference from flat space. On a larger scale, the violations are larger.

*

*Two longitude lines are parallel at the equator, but meet at the north pole.


*For a large square, you don't end up at the starting point. For example, start on the equator. Walk to the north pole and turn left $90^o$. The square will lead you back to the equator, along the equator, and back to the north pole.


*For a large triangle, the sum of the angles is $> 180^o$. Two longitude lines and the equator for triangles where the sum is anywhere between $180^o$ and $360^o$. By considering the equator to be three line segments, you can get a triangle where the sum is $540^o$.


*The distance between two points is given from latitudes and longitudes by the Haversine formula
$$d = 2\space r \space arcsin \bigg(\sqrt{sin^2\bigg(\frac{\phi_2 - \phi_1}{2} \bigg) + cos \space \phi_1 \cdot cos \space \phi_2 \cdot sin^2\bigg( \frac{\lambda_2 - \lambda_1}{2}\bigg) } \bigg)$$
If you start with $(x,y)$ coordinates and convert to $(\phi,\lambda)$, you will find the distance is different from flat space.

The surface of the earth curves by bending in a $3$rd dimension. It is mathematically possible to define a $2$D space where there is no $3$rd dimension. It is possible to define it so that geometry is distorted as you would expect for a sphere.
In general relativity, we work with a $4$D spacetime. We get no extra dimensions, but we do theoretically predict and observe distortions of geometry that match a curved spacetime. By this I mean we measure distortions in both distance and time intervals.
For example it is possible to show that time runs slower near the Earth than high above it. See Why can't I do this to get infinite energy?
You can define a square where one side is space and the other time. The square is high above the Earth. Two sides are separated by $1$ microsecond of time. Two sides are separated by $1$ light-microsecond of space in the direction toward Earth.
You can travel halfway around this square in both directions and find two different points for the opposite corner. It works like this.

*

*Start at an event high above the Earth. Wait $1$ microsecond, and find the event $1$ light-microsecond below you.

*Find the start event above the earth. Find the event $1$ light-microsecond below you. Go to that event and wait $1$ microsecond.

Because time runs slower near the Earth, you wind up at different events separated in time.
A: 
I can't understand the curvature of space because curvature of something happen in space while curvature of space need space itself to happen

What you are describing here is called extrinsic curvature. It is the kind of curvature that happens outside of a manifold. For example, this is the kind of curvature that a paper towel has while rolled up on a roll of paper towels.
That is not the kind of curvature that matters in general relativity. In GR we focus on intrinsic curvature, not extrinsic curvature.
Intrinsic curvature is inherent in the manifold itself. It does not need any external space to curve into. It simply means that the geometry within the manifold is not Euclidean. Triangles have more than 180 degrees, parallel lines can intersect, etc. There is no need of some higher dimensional embedding space.
A: I think maybe it is the terminology that causes the difficulty. You don't have to use the word "curvature". You can start from the following observations.
Observation 1. Define a straight line as a line of least distance between two points. If you make a closed shape in two dimensions using three straight lines, then in ordinary geometry (called Euclidean geometry) the internals angles of that shape (a triangle) add up to 180 degrees (that is, half a complete rotation).
Observation 2. Define a circle to be the set of points all at the same distance from some given point in two dimensions. Then in ordinary geometry (called Euclidean geometry) the circumferences of the circle will be equal to $\pi$ times its diameter, where $\pi$ can be expressed by various mathematical sums, and its value is approximately $3.14159265358979$.
Observation 3. If you make a shape out of three straight lines in our universe, somewhere near a massive body, then the internal angles of the shape will not add up to 180 degrees. The sum will differ from 180 degrees by an amount which is to do with the way the local gravitational acceleration is varying from place to place.
Observation 4. If you make a circle in our universe, somewhere near a massive body, then the circumference of the circle will not be equal to $\pi$ times its diameter. The ratio of circumference to diameter will differ from $\pi$ by an amount which is to do with the way the local gravitational acceleration is varying from place to place.
The term "curvature" is the standard term which is used to refer to observations 3 and 4 here, in comparison with observations 1 and 2. It is saying that geometry in our universe works differently from Euclidean geometry. General Relativity is a set of physical statements which include the assertion that this difference is to do with gravity, or, perhaps a better way to put it, gravity is the name we give to the result of this difference in geometry.
The reason why the word "curvature" is appropriate is because there is an easy way to visualize this change of geometry. A "curved" geometry in $N$ dimensions can be mathematically related to standard geometry in some larger number of dimensions, quite like the way a two-dimensional elastic membrane can take up interesting shapes in our three-dimensional space.
A: You can try and think of the curvature of space time as a geometric phenomenon.
Let's modify the popular rubber sheet visualization to see what that can mean:
Imagine our 4D space time as an (infinite) 2D rubber sheet. You can draw a grid of perfectly straight lines and right angles on that rubber sheet, but if you start pulling on the sheet at any point, or rather pinch it, what used to be perfectly straight lines become curved in the vincinity of the disturbance. Notice that you don't need an extra dimension to do so.
This example helps visualize part of the geometric aspects of curvature.
In our 3/4D space time photons/rays of light define straight lines through the vacuum. Any observed deviation from a straight line follows exactly the curvature of the space passed. It may be intuitive that the photon does not travel into or through another dimension when following a curvature; this may make clear that what should be a perfectly straight line, and from the perspective of the photon always is, becomes a curved line inside 3/4D space time.
A: Einstein suggested that matter curves space-time. So where there is matter, it will in any case distort space-time.
Maybe it's the other way around, space-time affects matter. Make matter spin in smaller and smaller circles until it is stuck in such a way that it slows down time and increases its entropy.
Imagine a tornado in the center of each galaxy.
What space-time is remains unknown, it has to move much faster than matter to blow through everything. Call it a kind of dark energy wind, which because of its unprecedented speed always drags matter to the smallest point of its vortex in space-time. Where ultimately only the heaviest elements can concentrate.
A: The "curvature" of space, or better, spacetime, is really better understood as a change in the laws of geometry in a region, which happens to obey the same mathematical rules as geometry done on a curved surface, and hence we "transfer" the concept of curvature from one situation to another. An analogy would be that there can be regions of space where that the inner angles of a suitably large triangle no longer add up to 180 degrees, but in other regions, for a triangle of the same size, they just might.
This analogy is not perfect, though, because of the "spacetime" part. This, in fact, forces more radical conclusions about the geometry of what, in our experience, look like extended objects. Generally, there is no "canonical" way to separate a curved spacetime into separate space and time parts, so we actually cannot talk of an "inherent spatial shape" of any object - a different set of coordinates, or "reference frame", will separate the two differently and tell us it has a different shape. What we can talk of, without requiring a convention, is their space-time shape, and it is the geometry of these - invisible! and four-dimensional - shapes that is altered by the curvature from one spacetime region to another.
A: Relativity theory is a theory that looks at the influence of gravity from a different perspective.
It's like every time anything passes a vacuum cleaner it will be sucked in, and an ant will think there is a hole or curvature in space at the vacuum hose end.
While in reality, the gravity mechanism is still an unsolved mystery!
